Cut steak diagonally across grain into thin slices.
Heat 1 teaspoon oil in a large Dutch oven over high heat. Add steak, and stir-fry 4 minutes or until browned. Remove steak from pan; set aside. Reduce heat to medium. Add remaining 1 teaspoon oil, shallots, and ginger, and stir-fry 1 to 2 minutes. Add water and next 6 ingredients. Bring to a boil; cover, reduce heat, and simmer 15 minutes.
While soup simmers, remove and discard stems from mushrooms. Slice mushroom caps into thin strips; set aside. Stir mushrooms, snow peas, and fish sauce into soup; cook 2 minutes or until peas are crisp-tender. Return beef to pan; cook 2 minutes or until thoroughly heated. Remove from heat; stir in basil. Discard cinnamon sticks and star anise. Ladle soup into bowls; top with bean sprouts and green onions. Serve with lime wedges.